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DYNAMICAL OCEANOGRAPHY
(B1) There can be a balance in (13.3) for D = δ a , i.e., λ V = δ D /D =
δ D /δ a ≪ 1 and we find Dρ/dt =0 . In this case, the density is constant along streamlines and hence the name of advective length scale for
δ a . In this advective limit, again (13.89) reduces to ρ = ρ s ; the solutions
of Dρ/dt =0, however, in general cannot satisfy this boundary condition
unless the Ekman pumping is negative.
(B2) Another balance is possible when D = δ D ≪ δ a such that λ V =1 .
In this case ρ has to be rescaled as ρ =( δ D /δ a ) 2 ˆ
ρ and consequently u
and v have to be rescaled as well according to the thermal wind balance.
Because δ D /δ a ≪ 1, (13.3) now becomes
w
∂ ˆ
ρ
∂z
=
∂ 2 ˆ
ρ
∂z 2 ,
(13.98a)
∂w
∂z
=0 .
(13.98b)
In the region with length scale δ D , the vertical velocity is constant in z and
equal to w E . Moreover, (13.98) only has bounded solutions for w E > 0
as is illustrated by Example 13.3.
◮
Example 13.3: Flow in a circumglobal channel
Consider the flow in a circumglobal channel (Fig. 13.7) that is bounded by
the latitudes θ = θ 0 and θ = θ 2 . At the surface, the flow is forced by a density
distribution ρ = ρ s (θ) and a wind-stress field τ φ = τ φ (θ) and τ θ =0. We assume
that the layer is infinitely deep and are searching for solutions that are independent
of the zonal coordinate, i.e., u(θ, z), v(θ, z), w(θ, z), p(θ, z) and ρ(θ, z).
θ = θ
0
θ = θ 2
τ
ρ s
Figure 13.7. Sketch of a circumglobal channel on the sphere (drawn here in the Southern Hemisphere).
Outside the Ekman layers the equations (13.3) become
v =0 ,
(13.99a)